Neural Extrapolation of Motion for a Ball Rolling Downan Inclined PlaneBarbara La Scaleia1, Francesco Lacquaniti1,2,3, Myrka Zago1*

1 Laboratory of Neuromotor Physiology, IRCCS Santa Lucia Foundation, Rome, Italy, 2 Department of Systems Medicine, University of Rome Tor Vergata, Rome, Italy,

3 Centre of Space Bio-medicine, University of Rome Tor Vergata, Rome, Italy

Abstract

It is known that humans tend to misjudge the kinematics of a target rolling down an inclined plane. Because visuomotorresponses are often more accurate and less prone to perceptual illusions than cognitive judgments, we asked the questionof how rolling motion is extrapolated for manual interception or drawing tasks. In three experiments a ball rolled down anincline with kinematics that differed as a function of the starting position (4 different positions) and slope (30u, 45u or 60u). InExperiment 1, participants had to punch the ball as it fell off the incline. In Experiment 2, the ball rolled down the incline butwas stopped at the end; participants were asked to imagine that the ball kept moving and to punch it. In Experiment 3, theball rolled down the incline and was stopped at the end; participants were asked to draw with the hand in air the trajectorythat would be described by the ball if it kept moving. We found that performance was most accurate when motion of theball was visible until interception and haptic feedback of hand-ball contact was available (Experiment 1). However, evenwhen participants punched an imaginary moving ball (Experiment 2) or drew in air the imaginary trajectory (Experiment 3),they were able to extrapolate to some extent global aspects of the target motion, including its path, speed and arrival time.We argue that the path and kinematics of a ball rolling down an incline can be extrapolated surprisingly well by the brainusing both visual information and internal models of target motion.

Citation: La Scaleia B, Lacquaniti F, Zago M (2014) Neural Extrapolation of Motion for a Ball Rolling Down an Inclined Plane. PLoS ONE 9(6): e99837. doi:10.1371/journal.pone.0099837

Editor: Ramesh Balasubramaniam, University of California, Merced, United States of America

Received March 11, 2014; Accepted May 12, 2014; Published June 18, 2014

Copyright: � 2014 La Scaleia et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permitsunrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: The authors confirm that all data underlying the findings are fully available without restriction. Zipped data of Experiment 1, Experiment 2,and Experiment 3 are available on Figshare under the DOI: http://dx.doi.org/10.6084/m9.figshare.1030546.

Funding: The work was supported by the Italian Health Ministry, Italian University Ministry (PRIN project), and Italian Space Agency (CRUSOE, COREA andARIANNA grants). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing Interests: The authors have declared that no competing interests exist.

* Email: m.zago@hsantalucia.it

Introduction

Humans are generally accurate in the manual interception of an

object dropped vertically from above, irrespective of whether the

task involves catching [1–3], punching [4–8] or batting [9,10].

The movements are time-locked to the expected arrival of the ball

over a range of different heights of fall [1,2,5]. Given that the

visuomotor delays for hand interceptions are considerable (100–

300 ms, see [3,11–13] and the visual estimates of image

acceleration are generally poor [11,14–17], the reported motor

accuracy depends on mechanisms extrapolating visual motion

forward in time by means of an internal model of gravity effects

[8,15,18–27].

Vertical free-falls represent the simplest case of gravitational

motion, because the gravitational acceleration is nearly constant

(g<9.8 m s22) on the Earth’s surface and the effects of non-

gravitational forces (such as air drag) can be neglected for many

interceptive actions [19,26]. It remains unclear how humans deal

with more complex gravitational kinematics. One experimental

strategy to address the problem of the predictive power of a model

of gravity effects is to follow Galileo and change systematically the

net acceleration due to gravity by means of an inclined plane [28].

For instance, an object slides down a frictionless plane with

acceleration equal to gsina, where a is the inclination angle relative

to the horizontal. If instead a homogeneous spherical object (such

as a ball) rolls down without slipping, its acceleration is (5/7)g(sina–

mvcosa/R), where mv is the coefficient of rolling resistance and R the

object radius (see Appendix). Are we able to cope with

accelerations that vary to such a large extent as a function of the

inclination angle and surface/object properties? Does the internal

model of gravity generalize to conditions with fractional gravity

effects?.

The answer to both questions would seem to be negative based

on studies which probe the explicit awareness of the effects of

gravity on sliding and rolling [29]. In such cases, perceptual

judgments tend to be systematically flawed. Bozzi [30,31] studied

the perception of sliding motion along a plane inclined at different

angles. He projected artificially generated motions of a square

target on a screen, and asked observers to choose the motion

function which looked like a natural, frictionless sliding down the

incline among several alternatives. He found that sliding is

perceived as natural not when it is uniformly accelerated, as

expected from physics, but when the object accelerates at the start

and then moves at constant speed for the rest. Hecht [32] used

computer-generated displays of wheels rolling down an inclined

plane to address the issue of whether observers are able to

appreciate the kinematic coupling of rotation and translation, and

the dynamic effects of gravity. He found that observers are unable

to differentiate between different acceleration functions, and their

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judgments are based on the translation component of the rolling

motion, while rotation is neglected. Vicario and Bressan [33]

described a related illusion: wheels appear to revolve much faster

than is compatible with their linear translation. Also the memory

of the final position for an object rolling down an inclined plane

without friction is distorted and the extent of the distortion

depends strongly on the slope [28]. Finally, people (adolescents)

asked to imagine a trolley freely rolling down along an incline,

once provided with information about total travel time, estimate

the traveled distance under the assumption of constant speed

motion, instead of accelerated motion [34].

According to Proffitt and Gilden [35], people make judgments

about natural object motions on the basis of only one parameter of

information that is salient in the event, and encounter difficulties

when evaluating the dynamics of any mechanical system that has

more than one dynamically relevant parameter. This accounts for

the very poor understanding of wheel dynamics [36]. Notice,

though, that the visual system can use contextual cues, along with

intrinsic surface cues, to compute percepts of rolling objects [37].

On the other hand, it is well known that automatic visuomotor

responses generally are much more accurate and less prone to

perceptual illusions than cognitive judgments about the same

targets, consistent with the idea that visual information for action

and for explicit awareness are processed in a distinct manner (e.g.,

[38–40]). Thus, perceived slant of an incline is grossly overesti-

mated, whereas the corresponding action measures are accurate

[41,42]. Moreover, rolling objects with rotational and translational

motion congruent with an object rolling on the ground elicit faster

tracking eye movements during pursuit initiation than incongruent

stimuli [43].

To our knowledge, only one study dealt with the manual

interception of an object shifting along slanted trajectories, and it

relied on simulated motion in a virtual reality setup [44]. In this

study, participants were asked to abduct their index finger at the

time of arrival of a ball rolling within a tube which could have

various shapes. On average, participants responded too early and

their errors varied with the slope of the tube, indicating that

interception was not accurate. Moreover, the errors were similar

when the tube was invisible depriving subjects of any cue that

could be used to predict the target trajectory in advance, and when

the visual scene was projected upside down violating ecological

gravity constraints. This indicates that the task could be

accomplished, to a certain degree, by visual online feedback only.

However, the errors in catch trials, in which the speed of the ball

was unexpectedly maintained constant, differed depending on

whether the tube was visible or not. This pattern of errors

suggested that participants learned to predict some effects of

acceleration normally induced by the shape of the tube within the

practice period. It should be noticed that, because the virtual scene

did not include any scaling or texture cues, the dynamic

parameters (gravity, mass, friction) were undetermined, and this

might explain the presence of both bias and variability in the

performance, as well as the heavy reliance on visual online

feedback [44].

We examined the issue by investigating the interception of a real

object rolling down an inclined plane. Both visual perception and

motor interaction involving real objects can differ substantially

from those involving virtual objects [45–47]. In particular,

interception of a target falling vertically [6] or swinging as a

pendulum [45] under gravity is more accurate with a real ball than

with a virtual target. Moreover, the role of internalized laws of

dynamics may become more relevant in natural environments

complying with the ecological constraints of the real world [48].

Here we performed three experiments in which a ball rolled

down an incline with kinematics that differed as a function of the

starting position and slope. In Experiment 1, participants were

asked to punch the ball after its exit from the incline. In

Experiment 2, the ball rolled down the incline but was stopped at

the end; participants were asked to punch the imaginary ball as if it

continued its motion. In Experiment 3, the ball rolled down the

incline and was stopped at the end, as in Experiment 2, but here

participants were asked to draw with the hand in air the trajectory

that would be described by the ball if it kept moving. By varying

the motor task and/or the feedback about the terminal part of ball

trajectory across experiments, we aimed at elucidating the nature

of the predictive processes involved in the extrapolation of the ball

trajectories.

Experiment 1

In this experiment, participants intercepted a ball falling from

an inclined plane. We changed the slope a of the incline in

different blocks of trials, and randomized the starting position of

the ball on the incline (resulting in different durations of ball

motion) across trials. Accordingly, ball acceleration on the incline

and speed at the time of interception varied by about two times

over the range of conditions. To impose stringent margins of

spatial and temporal accuracy, we constrained the valid intercep-

tion region to a small volume along the path of fall of the ball. In

this manner, we were also able to match the duration of ball

motion prior to interception across all tilt angles.

Different hypotheses about the law of ball motion which is

assumed by the subjects make different predictions about the

timing of the interceptive responses. If subjects assumed constant

speed motion, they would arrive too late at the interception point

and miss the ball. If instead they assumed constant acceleration

equal to Earth gravity for all tilt angles, they would arrive too

early. They would also arrive too early if they took into account

the translational but not the rotational component of ball motion.

Only if all components of rolling motion were taken into account,

would interceptions be timed correctly independent of tilt angle.

MethodsParticipants. All participants in this and the following

experiments gave written informed consent to procedures

approved by the Institutional Review Board of Santa Lucia

Foundation, in conformity with the Declaration of Helsinki on the

use of human subjects in research. All participants were right-

handed (as assessed by a short questionnaire based on the

Edinburgh scale), had normal or corrected-to-normal vision, no

past history of psychiatric or neurological diseases, and were naıve

to the specific purpose of the experiments. Fifteen subjects (11

females and 4 males, 2867 years old, mean 6 SD) participated in

Experiment 1.

Experimental set-up. Subjects sat on a chair placed in front

of a vertical projection screen (in the following denoted as

‘‘screen’’; Fig. 1), and in front and to the right of the inclined

surface (in the following denoted as ‘‘incline’’) from which a ball

rolled down (Fig. S1). The screen was 0.45 m wide and 1.4 m

high. The incline was 2.1 m long and 0.2 m wide. Its surface was

made of inox steel, and served as the track for the rolling ball. The

surface had been especially machined so as to be very smooth (see

Appendix for estimates of dynamic friction). It was glued as a

groove to a 6-cm thick laminated poplar, and the whole structure

was supported by three tripod stands. The longitudinal axis of the

incline and thus the direction of ball descent were roughly parallel

to the frontal plane of the subject. By changing the height of the

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tripods and the orientation of their joints, the incline could be

tilted relative to the horizontal by one of 3 angles (30u, 45u, or 60u)in different blocks of trials. Irrespective of the tilt angle, the lower

end of the incline and exit point of the ball were at 0.81 m height

above the floor. A soft, homogenous rubber ball (diameter, 9 cm;

weight, 30.2 g) rolled down the incline with different accelerations

depending on the starting position and tilt angle, without slipping

or bouncing. After exiting from the lower end of the incline, the

ball fell under gravity and air drag along a quasi-parabolic

trajectory. Notice that the homogeneous surface of the balls

provided minimal optical cues about the rotational component of

the motion.

The incline was equipped with 12 electromechanical lever arms

spaced along the track, which allowed the release of the ball from

different starting positions (see Protocol). These devices were

located on the side of the incline opposite to that facing the subject;

consequently, they did not prevent the view of the descending ball

nor did they interfere with ball motion. Each lever consisted of a

thin steel rod whose position (up or down) was rapidly set by a

solenoid (G.W. Linsk, Clifton Springs, NY) under computer

control. The time accuracy of ball release was better than 1 ms.

An on-axis laser transmitter/receiver (transceiver) was placed at

5 cm distance from the lower end of the incline, orthogonal to the

direction of ball descent, to monitor the descent time of the ball in

each trial.

The back of the chair, supporting the head and torso of the

subject, was adjusted at about 25u relative to the vertical, allowing

a comfortable position and full view of the incline. Subject’s eyes

were at a horizontal distance of about 50 cm from the incline

longitudinal axis, 8 cm to the right and 31 cm above the lower end

of the incline. Their horizontal distance from the screen was about

1.2 m. By protracting the arm forwards, the subject could easily

reach the interception region just below and to the right of the

lower end of the incline.

Subjects held an instrumented plastic box (size: 266.563.2 cm

[w6h6d]; weight: 65 g) in their right hand. The box had a

protruding steel pin (4.2 cm length, 0.4 cm diameter) on the front

side, and a steel winged frame fixed over the horizontal side. The

box was grasped by the subject so that the pin (in the following

denoted as ‘‘hitter’’) protruded between the index and middle

finger, and the winged frame was roughly parallel to the back of

the hand (Fig. 1, inset). Inside the box there was a tri-axial

piezoelectric accelerometer which measured box (and hand)

acceleration (Isotron 63-B100, Endevco, San Juan Capistrano,

CA; 650 g dynamic range, amplitude nonlinearity ,1%). The

accelerometer and the laser receiver were sampled by the

acquisition system at 1 kHz. The position of the box in 3D was

recorded at 200 Hz by means of the Optotrak 3020 system

(Northern Digital, Waterloo, Ontario; 63 SD-accuracy better

than 0.2 mm for x, y, z coordinates). To this end, three infrared

emitting markers were attached to the winged frame of the box.

The 3D position of the tip of the hitter was derived from the

known geometry of the box plus hitter, and the measured 3D

position and orientation of the box. Three additional markers were

placed on the incline to determine its position and orientation, one

at 80 cm and the other two at 1 cm distance from the lower end of

the incline.

Synchronous acquisition of all motion data (accelerometer,

Optotrak, and laser receiver) was accomplished by the real-time

system PXI-1010 (National Instruments, Austin, Texas) pro-

grammed with custom-made software. In each trial, data

acquisition was started by the experimenter about 250 ms before

ball release and lasted 2 s.

Task. Subjects were asked to punch the ball falling from the

incline with the hitter. The ball had to be hit at the time its center

reached the nominal interception point (nIP) along the quasi-

parabolic path. nIP was located 5.5 cm below, and 13.5, 11 or

8 cm to the right of the lower end of the incline, for tilt angles of

30u, 45u or 60u, respectively. We provided continuous feedback of

Figure 1. Schematic of the experimental setup. For simplicity, only one tripod stand is depicted (instead of the 3 tripods actually used).doi:10.1371/journal.pone.0099837.g001

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task variables by projecting on the screen in front of the subject 1)

the time-varying position of the hitter, 2) the nominal starting

position of the hitter (nSP) fixed for each trial, 3) the nominal

interception point (nIP) fixed for each trial, and 4) the position of

the lower end of the incline. Two 2D-projections with a front-view

and a top-view of these points were shown side by side on the

screen. The position of the box with the hitter was updated in real

time (within 0.5 ms) by means of the acquired Optotrak data. nSP

and nIP were shown inscribed within a 2 cm side square (green for

nSP and red for nIP).

Subjects kept a relaxed posture between trials. Shortly before

trial start, they were asked to look at the starting position of the

ball on the incline and at the corresponding nIP. They were

prompted to recoil the arm in the starting posture so as to place

the tip of the hitter within the 2-cm-square around nSP. With the

adducted shoulder, the upper arm was roughly vertical, the

forearm horizontal, the wrist mid-pronated, the hand and fingers

clenched around the box. In this way, the nSP was located

12.5 cm below and 13.5, 11 or 8 cm to the right of the lower

border of the incline, for tilt angles of 30u, 45u or 60u, respectively.

The initial horizontal distance between the nominal starting

position of the hitter tip and the frontal plane passing through the

longitudinal axis of the incline was 20 cm.

After a pseudo-random delay of 100, 200, 300 or 400 ms, the

ball was released and rolled down the incline with the kinematics

described in the Appendix. Apart from the on-line feedback on

the screen and the direct sensory information about ball trajectory,

no further information was provided to the participants about

their performance in hitting the ball.

Protocol. In separate blocks of trials, the incline was tilted by

30u, 45u, or 60u (order counterbalanced across participants). Ball

acceleration was roughly constant on the plane for a given

inclination, being 3.21 m?s22, 4.71 m?s22 and 5.90 m?s22 for tilt

angles of to 30u, 45u and 60u, respectively, while it was 9.81 m?s22

during the free-fall phase of all tilts (see Appendix). At each tilt

angle, there were 4 possible starting points of the ball along the

incline (different among tilt angles), resulting in 4 nominal

durations of ball motion (nBMD, see Table S1 and Fig. S1).

nBMD was the total motion duration from release to nIP and was

matched across tilt angles. nBMD was randomized across trials,

avoiding identical values in consecutive trials. In a block of trials,

each nBMD was presented 15 times for a total of 60 trials at each

tilt angle. Thus, one experiment included 180 trials (60 trials63 tilt

angles). Participants were allowed to pause any time they wished

during the experiment. Total duration of an experiment was about

2 hours.

Data analysis. Out of a total of 2700 trials (180 trials615

subjects), 60 were excluded (2.2%) from the analysis due to the

presence of artifacts or lack of subject’s attention (as marked in the

experiment’s notebook). When the latter event occurred, the

subject was allowed to pause briefly so as to recover full attention

to the task. Except for the detection of contact events (see below),

raw kinematic data were numerically low-pass filtered (bi-

directional, 25-Hz cutoff, 2nd order Butterworth filter) to eliminate

high-frequency oscillations due to contact. The instantaneous

velocity of the hitter was estimated by numerically differentiating

(finite difference approximations) the recorded x, y, z coordinates.

Endpoint analysis. We evaluated the position of the hitter when it

first reached the minimum distance from the nominal interception

point nIP. This position is denoted interception point (IP) in the

following.

Timing error. For each trial, we computed the timing error (TE)

as the difference between the time of arrival of the hitter in IP and

nBMD. A positive value of TE corresponds to a response later

than that theoretically expected (and a negative value to an earlier

response) if the hitter arrived at IP at the same time as the center of

the ball. Mean TE and 95% confidence interval were computed

over all trials of each condition (tilt angle and nBMD). The

theoretical margin of error for the timing of successful hitting

movements through nIP was on the order of625 ms, depending

on the velocity of the ball. In fact, the actual margin was larger

because the ball could be hit at any point on its surface and

different points along the descent.

To characterize the rate at which subjects improved their timing

during an experiment, we fitted an exponential function to the

series of TE values across successive trials (non-linear least-squares

fitting):

TEi~b0zb1 exp {i=b2ð Þ ð1Þ

where TEi is the TE value for repetition i, while b0, b1, and b2 are

the offset, gain, and learning-constant.

As an additional estimate of the timing errors, we used a

spatiotemporal criterion that takes into account the physical

dimensions of the ball. Accordingly, we considered the time at

which the hitter first arrived at the minimum distance from the

surface of the ball with its center in nIP, and compared it with the

actual arrival time of the ball in the same position.

Hitter position. For each trial, we computed the spatial error as

the Euclidean distance between IP values and nIP. Mean spatial

error and 95% confidence interval were computed over all trials of

each condition (tilt angle and nBMD). In addition, we computed

the 2D spatial distribution of IP in the plane of the ball’s flight for

all trials of each condition.

For a given tilt angle a and nominal ball motion duration

nBMD, let nIPa,nBMD denote the nominal interception point and

IPa,nBMDs,j the position, for trial j of subject s, of the hitter-tip when

it first reached the minimum distance relative to the nominal

interception point nIP (i.e. at interception time). At interception

point IP, the systematic spatial bias SBa,nBMDs of subject s is

characterized in terms of the 2D vector difference between the

hitter-tip position at interception (averaged across all repetitions of

nBMD) and nominal interception pointNIPa,nBMD:

SBa,nBMDs ~IPa,nBMD

s {nIPa,nBMD ð2Þ

The values obtained in all individual subjects were averaged to

yield the overall systematic spatial biasSBa,nBMD.

The deviation d from the mean for a given trial j of subject s is

defined as

da,nBMDs,j ~IPa,nBMD

s,j {IPa,nBMDs ð3Þ

The 262 matrix of variance and covariance (hereafter referred

to as the covariance matrix) of subject s is expressed by the

equation:

Sa,nBMDs ~

Xjda,nBMD

s,j da,nBMDs,j

� �T

nta,nBMDs {1

ð4Þ

where nta,nBMDs represents the number of trials of subject s for

angle a and nominal ball motion duration nBMD and T denotes

the transpose.

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By pooling the interception points across all subjects, the

estimate of the combined covariance matrix for angle a and

nBMD is:

Sa,nBMD~

Xs

Xjda,nBMD

s,j da,nBMDs,j

� �T

Xsnta,nBMD

s {nsð5Þ

where ns~nsa,nBMD is the number of subjects (ns = 15).

The spatial precision (variability of performance) of all subjects

for tilt angle a and nBMD was computed by means of the 95%

tolerance ellipse for each condition. This ellipse, based on the data

from all trials of all subjects for a given condition, is obtained by

scaling the combined covariance matrix:

Ta,nBMD0:95 ~

p nz1ð Þ n{nsð Þn n{ns{pz1ð ÞF

a,nBMD0:05,p,n{ns{pz1Sa,nBMD ð6Þ

where n~X

snta,nBMD

s is the total number of trials of all subjects

for a given condition, and p ( = 2) is the dimensionality of the

Cartesian vector space [49].

Eigenvalues and eigenvectors of the Ta,nBMD0:95 matrix were used

to characterize the shape, size and orientation of the ellipse

[50,51]. The two axes of the ellipse have the same orientation as

the two eigenvectors of Ta,nBMD0:95 matrix, and the lengths of the two

semi-axes correspond to the square roots of the two corresponding

eigenvalues. We tested whether the eigenvalues were statistically

different between each other by means of a x2 test at 95%

confidence level x2wx2

0:05,2

� �, where x2 has the form ([51], p.

336):

x2~{ n{1ð ÞX

kln la,nBMD

k z n{1ð Þr ln

Xk

la,nBMDk

rð7Þ

wherela,nBMDk are the r~2 eigenvalues being compared for a given

covariance matrix Ta,nBMD0:95 k~1{2ð Þ. x2 has q~0:5r rzð

1Þ{1~2 degrees of freedom.

Contact. To analyze the contact between the ball and the hitter

(or hand/box), the raw data from the accelerometer were filtered

with a bidirectional 25-Hz high-pass Butterworth filter. Visual

inspection of the filtered accelerometer traces in single trials

allowed to recognize contact by the presence of a brief burst of

high-frequency oscillations [7]. Contact time denotes the time of

occurrence of these oscillations. Detailed visual inspection of the

combined data set of hitter position (measured with Optotrak) and

acceleration (measured with the accelerometer) showed that, when

the hitter passed close to the nIP, contact oscillations almost

invariably started within the time interval between the positive

peak and the negative peak of acceleration, consistent with

previous work on ball punching [6,7]. The contact oscillations

started outside this interval in only 3.38% of all trials. Accordingly,

we assigned a score of 1 to trials in which the contact oscillations

started within the time interval defined above, and a score of 0 if

there were no detectable oscillations or they started outside that

interval. The contact rate was then computed as the percentage of

all trials scoring 1 over the total number of trials of all subjects.

Hitting movement. We estimated the onset time of hitter movement

(MOT, time when the hitter-tip speed first reached 0.05 m?s21)

and movement duration MD (interval between MOT and hitter

arrival time in IP).

Tangential velocity (denoted as ‘‘speed’’ in the following) of the

hitter was computed as vT~

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi_xx2z _yy2z _zz2

q. Initial hitter speed

(ISpeed) was defined as the average speed of the hitter in a window

of 100 ms starting from MOT. We measured also the peak of

hitter speed (PSpeed) during the hitting movement and the hitter

speed in IP (Speed_IP). The curvature of hitter trajectory (TCurv)

was defined as the mean of the perpendicular distances of all

sampled points along the hitter trajectory (from movement onset to

IP) relative to a straight line connecting the position of the hitter at

movement onset and IP [52].

Statistics. Differences between conditions were assessed

using Repeated Measures ANOVA (RM-ANOVA with tilt angle,

nBMD and repetition as within-subjects factors) with post-hoc

Bonferroni corrections for multiple comparisons (P,0.05).

Results and DiscussionInterception performance. Interception timing was often

close to ideal (i.e., timing error TE = 0). The mean TE (computed

over all repetitions of all conditions) was 16 ms (SD = 26 ms,

n = 2640, Fig. 2A), within the theoretical margin of error (see

Methods). Three-way RM-ANOVA (3 tilt angles64 nBMDs615

repetitions) showed that TE depended significantly on the nominal

duration of ball motion nBMD (F3,42 = 19.91, P,0.001) and

repetitions (F14,196 = 4.321, P,0.001), while it was independent of

tilt angle and any interaction between factors (P.0.092). In

particular, TE tended to decrease with increasing nBMD and

repetition. Post-hoc tests showed that TE for the longest duration

(nBMD = 730 ms) was significantly smaller than that for the other

nBMDs, while TE for the first repetition was significantly higher

than that in all subsequent repetitions (all P,0.05) except

repetition 2, 4, and 6.

Neither the mean TE computed over the first repetition of all

conditions nor the mean TE computed over the first 3 repetitions

of all conditions depended significantly on tilt angle, nBMD or

interactions (all P.0.1). The effect of practice on timing was

quantified by best-fitting (r2 = 0.731) eq. 1 to the TE of all subjects

and conditions as a function of repetition (Fig. 3). We found a short

learning constant (b2 = 2.532, 95% confidence limits: [0.5104,

4.563]), indicating that learning occurred over the first few

repetitions: TE was reduced by 95% of the overall change relative

to the steady state value after only 7 repetitions. The other fitting

parameters were: offset b0 = 0.01383 ms (95% confidence limits:

[0.01205, 0.0156] ms) and gain b1 = 0.0106 ms (95% confidence

limits: [0.0069, 0.0143] ms).

Qualitatively similar results were obtained by using an

alternative estimate of the timing errors that takes into account

the physical dimensions of the ball (see Methods). The mean

difference between the time at which the hitter was closest to the

position of the ball surface and the actual arrival time of the ball,

computed over the last 8 repetitions (steady-state) of each

condition, was 1621 ms, 6623 ms and 4620 ms for tilt angles

equal to 30u, 45u and 60u respectively. These timing errors are

even smaller than the TE values reported above, and confirm the

high spatiotemporal accuracy of the interceptive actions at steady-

state.

In agreement with this conclusion, we also found that the mean

spatial error (distance between IPs and nIP) was 3.90 cm

(SD = 1.93 cm, n = 2640), smaller than the ball radius (4.5 cm).

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Spatial error (Fig. 4A) depended slightly but significantly on tilt

angle (3-way RM-ANOVA, 3 angles64 nBMDs615 repetitions

(F2,28 = 4.786, P = 0.016), while it was independent of nBMD,

repetitions, and any interactions (P.0.093). Post-hoc tests revealed

that the spatial error was significantly smaller for 30u than 45uangle (P = 0.02).

The scatter of interception points across trials was not simply

due to random variability, but was accounted for, at least in part,

by the attempt of the subjects to intersect the ball trajectory at a

point placed along the path of the ball. This is demonstrated by

the statistical analysis of the spatial distribution of IPs in single

trials. To this end, we computed the tolerance ellipses including

95% of IPs (T0.95) for each experimental condition (n = 12, 4

nBMD63 angles). These ellipses (red ovoids in Fig. 5) are

projected in the plane of motion of the ball’s center and are

centered on the mean IP for any given condition. The magnitude

of the variable error is provided by the square root of the

eigenvalues of T0.95 (see Methods). We found that 11 out of 12

ellipses had statistically different eigenvalues (Table S2), implying

that the distribution of positions was not isotropic but tended to

align along specific directions. Critically, the 95% confidence

limits of the first eigenvector (aligned with the major axis of the

ellipse, Table S3) included the tangent in the mean IP to the

trajectory of the ball in several conditions: all nBMDs at 30u tilt

angle (although the two eigenvalues were not significantly different

for nBMD = 730 ms, 30u tilt), the 3 shortest nBMDs at 45u, and

the shortest nBMD at 60u. In all such conditions, the individual

responses tended to be distributed along the ball trajectory,

consistent with the hypothesis that subjects were able to predict the

Figure 2. Timing error TE for each condition (tilt angle and duration of ball motion nBMD). Mean695% confidence interval, over allsubjects and repetitions (n = 225, except for missing trials, see text). A) Results from Experiment 1. B) Results from Experiment 2.doi:10.1371/journal.pone.0099837.g002

Figure 3. Experiment 1. Effect of practice on timing errors. Mean TE(695% confidence interval, over all subjects, tilt angles and nBMD,n = 180) is plotted for each repetition (red), together with theexponential best-fit (black), and the 95% decrement (dotted gray).doi:10.1371/journal.pone.0099837.g003

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trajectory. In this regards, notice that the tolerance ellipses overlap

appreciably more with the parabolic trajectories than with the

linear trajectories which would be followed by the ball if it

continued rolling on an extended inclined plane for 30u and 45utilt (Fig. 5). For instance, the major axis of the ellipse for 30u tilt,

nBMD = 610 ms and for 45u tilt, nBMD = 730 ms deviated by

about 15u from the linear path. Instead, the ellipses overlap with

both parabolic and linear trajectories for 60u tilt, because these two

sets of trajectories diverge further down at 60u-tilt.The area of the ellipses increased significantly (r2 = 0.849) with

the speed of the ball at nIP (Fig. 6A), indicating that, in different

trials, subjects intercepted faster balls at points distributed along a

greater path stretch, consistent with the fact that the ball covered a

longer path segment in the unit time with higher speeds. The slope

of the linear regression was 0.0034 m?s (95% confidence limits:

[0.0024, 0.0044]), the intercept was 0.0032 m2 (95% confidence

limits: [25?1025, 0.0063]).

A different way of assessing the subjects’ ability to predict ball

trajectory in single trials is provided by the analysis presented in

Fig. 7 (red dots). Here we computed the point of the first

intersection of the hitter trajectory with the plane of movement of

the ball’s center (hitter intersection point), the time at which this

occurred (hitter intersection time), the position of the ball when it

first reached the minimum distance from the intersection point,

and the time at which this occurred (ball intersection time). We

then performed a linear regression (red line) between ball

intersection time and hitter intersection time. If indeed subjects

anticipated ball trajectory, the values of ball intersection time in

single trials should covary with the corresponding values of

intersection time. This expectation was borne out from the results

shown in Fig. 7. For all experimental conditions, the correlation

coefficient was significantly higher than zero (P,0.001), the slope

of the regression averaged across condition was 0.35, and the

average intercept was 0.41. Consistent with the small spatio-

temporal errors reported above, we found that the mean contact

rate was very high (87.5561.4%, 3 angles64 nBMDs615 subjects,

n = 180), implying that subjects were very often successful at

hitting the ball close to the nIP (see Methods).

Movement characteristics. Typical paths of the hand are

plotted in red in Fig. 8, and average movement parameters are

presented in Table S4. In general, the hitting movements followed

a curved path from an almost invariant starting position (dictated

by nSP) to a more variable position close to the interception point.

In the previous section we have shown that this spatial variability

near interception can be accounted, at least in part, by the attempt

of the subjects to intersect the ball trajectory at points distributed

along the path of the falling ball.

Path curvature TCurv was quantified in terms of the mean

distance of the hitter path from a straight line. Average TCurv was

17.16 mm (SD = 9.01 mm, n = 2640). It depended significantly on

nBMDs (3-way RM-ANOVA, 3 angles 64 nBMDs 615

repetitions, F3,42 = 1.1945, P = 0.011, the shorter the nBMD, the

straighter the hitter trajectory), but did not depend significantly on

the other factors and their interaction (P.0.085).

A significant dependence on nBMDs was also shown by arm

movement duration (MD, F3,42 = 3.706, P = 0.019), maximum

Figure 4. Spatial error (mean±95% confidence interval, over all subjects and repetitions, n = 225) for each condition (tilt angle andnBMD). A) Results from Experiment 1. B) Results from Experiment 2.doi:10.1371/journal.pone.0099837.g004

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speed (PSpeed, F3,42 = 8.528, P,0.001), and speed at interception

(Speed_IP, F3,42 = 6.509, P,0.001). Thus, the longer the nBMD,

the longer was MD, the higher PSpeed and Speed_IP (see Table

S4). Both PSpeed and Speed_IP also depended significantly on

repetitions (F14,196 = 3.779, P,0.001 and F14,196 = 4.549, P,0.001

respectively), increasing with practice. PSpeed depended also on

the interaction between nBMD and repetition (F42, 588 = 1.6707,

P = 0.00603). On the other hand, the initial speed (ISpeed) did not

depend significantly on any factor or interaction (P.0.127). The

observation that ball motions of higher final speed (longer nBMD)

were associated with faster arm movements agrees with several

previous reports on the interception of targets with variable speeds

under visual guidance [11,53,54].

Typical speed profiles are plotted in Fig. 9 for a representative

subject. It can be noticed that the profile for the first repetition

(red) does not overlap with the average profile (blue) computed

over the repetitions from 4th to 15th of the same condition, but is

anticipated in time. In addition, while the average profile exhibits

the typical bell-shaped waveform of fast reaching movements with

little, if any, on-line corrections [55], the profile of the first

repetition is biphasic. Hand speed reached a first peak and then

decreased about 100 ms prior to the nominal interception time, to

peak again around interception time. This biphasic profile was

observed in the very first trial of all subjects, and more sporadically

in the first repetition of the other conditions. When the profile of

the first repetition was not obviously biphasic, the rise time of

speed was nevertheless slower than in the following repetitions.

The presence of submovements in the tangential speed profile may

be indicative of on-line corrections, possibly based on visual

feedback.

In sum, the average interception position and time were very

close to those required by the instructions, and this was so at all tilt

angles and starting positions of the ball on the incline. Individual

interception positions in single trials tended to be distributed along

the trajectory of ball motion, consistent with the hypothesis that

subjects were able to fully extrapolate the trajectory. There was

also clear evidence for rapid learning: timing errors were reduced

and movements became more stereotyped after the first few

repetitions of practice. Also, balls arriving at higher speed were

intercepted more vigorously, with faster movements. A similar

covariation between arm speed and target speed has previously

been shown to result from continuous adjustment of interceptive

movements based on visual feedback of target speed with a delay

of about 200 ms [11].

The findings from Experiment 1 are compatible with the

hypothesis that participants took into account both the transla-

tional and rotational component of rolling motion of the ball on

the incline, as well as the subsequent free-fall kinematics. As a

result, they were able to deal with widely different accelerations on

the incline and final speeds at interception over the full range of

conditions.

Experiment 2

The previous experiment showed that, with continuous visual

feedback plus haptic feedback of hand-ball contact in successful

trials, subjects are able to extrapolate accurately the gravitational

motion of a real target with very different kinematics. We now

address the question of whether extrapolation is still possible when

there is no visual feedback about target kinematics during the

critical terminal part of the trajectory, nor haptic feedback of

hand-ball contact.

In Experiment 2, the ball rolled along the incline as in

Experiment 1, but it was stopped just before the end of the track.

Participants were asked to imagine that the ball kept moving, and

to punch it with the hitter at the same nominal interception point

Figure 5. Experiment 1. Spatial distribution of interception points. IPs in single trials (black dots) and 95%-tolerance ellipses (red) for eachcondition. The first eigenvector (along the major axis of the ellipse) with the 95%-confidence cone (blue straight lines) is drawn when significant (seetext). The gray area within black curved lines represents the envelope of the ball trajectory, and the blue curved line is the trajectory of the ball center.The pink straight lines represent the linear trajectory that would be followed by the ball if it continued the motion on the extended inclined plane.doi:10.1371/journal.pone.0099837.g005

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as in Experiment 1. It should be remarked that, because the

unseen portion of ball trajectories mainly corresponded to a free-

fall along quasi-parabolic paths, a simple linear extrapolation of

the visible portion would not allow a correct prediction of the

interception point and time, at least for 30u and 45u tilts (see Fig. 5).

MethodsParticipants. Fifteen subjects (11 females and 4 males, 2867

years old) participated in Experiment 2. Seven of them had

previously participated in Experiment 1 (about 5 months before).General procedures. The experimental setup, task and

protocol were similar to those of Experiment 1, with the following

notable changes. In this experiment, the ball rolled along the

incline until it was stopped by a rod placed 2 cm before the end of

the track, and remained there for the rest of the trial. Participants

were asked to imagine that the ball kept moving beyond the stop.

They had to punch the imaginary ball with the hitter at the same

nominal interception point (nIP) as in Experiment 1. Notice that

they never saw the ball falling off the incline during this

experiment.

Data analysis was identical to that of Experiment 1, except that

contact rate could not be assessed, given that there was no physical

interaction with the ball. Out of a total of 2700 trials (180

trials615 subjects), 39 were excluded (1.4%) from the analysis due

to the presence of artifacts or lack of subject’s attention. Because

we did not find any systematic difference between the group of 7

subjects who had previously performed Experiment 1 and the

group of 8 subjects who did not, the results from the two groups

were pooled together.

Results and DiscussionInterception performance. Despite the virtual nature of the

interception which lacked visual and haptic feedback of a hit ball,

both the temporal and spatial errors were limited. The mean

timing error TE (computed over all repetitions of all conditions)

was 213 ms (SD = 69 ms, n = 2661, Fig. 2B), within the

theoretical margin of error. However, there was a gradient of

the response timing with increasing tilt angle: on average, TE was

251667 ms, 3657 ms and 15668 ms for 30u, 45u and 60u,respectively. Three-way RM-ANOVA (3 angles64 nBMDs615

repetitions) showed that TE depended significantly on angle

(F2,28 = 25.76, P,0.001), whereas it was independent of nBMD,

repetition and any interactions (P.0.062). Post-hoc tests showed

that TE for 30u was significantly different from that for the other

angles (P,0.001).

This trend is not compatible with the possibility that subjects

extrapolated the visible trajectories under the assumption that the

ball continued rolling on an extended inclined plane. If this were

the case, in fact, the responses should be late (because rolling is

slower than free-fall), with the longest time delay for 30u tilt and

smaller delays with increasing tilt angle, the opposite trend of what

we found.

A possible explanation for the early responses at 30u tilt is that

subjects misperceived this tilt angle. It is known that observers tend

to overestimate the slant of an incline [42,56], more so for smaller

than larger tilts [57]. Such a differential estimate of tilt angle might

account for the observed gradient of responses, because larger

overestimates of 30u tilt would predict earlier responses than for

45u or 60u tilts. However, it has been shown that the estimates of

an incline tilt are generally correct when an action is performed on

the incline [42,57].

A more plausible explanation is that the gradient of responses as

a function of tilt angle is related to a central tendency effect. The

ball motion that subjects had to imagine in order to fill in the

occluded gap (that is, the time gap between ball stop on the incline

and the theoretical ball arrival at nIP) lasted less and less with

increasing tilt angle (see Table S1). If subjects assumed that the

occluded duration of fall was the same for all angles and equal to

that for 45u angle, they would arrive at nIP early for 30u, on time

for 45u, and late for 60u. Indeed, the difference between mean TE

for 30u and that for 45u (54 ms) roughly corresponds to the mean

difference in duration of fall for these two angles (38 ms), and the

difference between mean TE for 45u and 60u (12 ms) roughly

matches the corresponding difference in duration of fall (15 ms).

This observation suggests that subjects used an interceptive

strategy based on a global assessment of the unseen ball kinematics

across all tested conditions. In this regards, it should be recalled

that tilt angle was blocked (although in counterbalanced order

across subjects), while nBMD was randomized across trials.

The overall mean spatial error (distance between IPs and nIP)

was 4 cm (SD = 2 cm, n = 2661), essentially identical to that of

Experiment 1. Spatial error (Fig. 4B) depended slightly but

significantly on tilt angle (3-way RM-ANOVA, 3 angles64

nBMDs615 repetitions (F2,28 = 6.54, P,0.005) and repetitions

(F14,196 = 3.997, P,0.001), while it was independent of nBMD and

any interactions (P.0.178). Post-hoc tests revealed that the spatial

error was significantly smaller at 30u than 60u angle (P,0.005),

while it was significantly greater in the first repetition than all other

repetitions (except repetitions 2 and 5).

Figure 6. Area of the 95%-tolerance ellipses as a function ofball speed at nIP: data points (red and blue) and linearregression (black). A) Results from Experiment 1. B) Results fromExperiment 2.doi:10.1371/journal.pone.0099837.g006

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The effect of practice on the spatial error was quantified by best-

fitting (r2 = 0.721) eq. 1 to the spatial error of all subjects as a

function of repetition (Fig. 10). We found a short learning constant

(b2 = 2.095, 95% confidence limits: [0.4363, 3.753]), so that the

error was reduced by 95% of the overall change relative to the

steady state value after 6 repetitions. The other fitting parameters

were: offset b0 = 0.03871 m (95% confidence limits: [0.03753,

0.0399] m) and gain b1 = 0.008 m (95% confidence limits:

[0.005095, 0.0109] m).

Although the errors were limited, in contrast with Experiment 1

the spatial scatter of individual reaches was not correlated with the

virtual path of the ball from the incline until the nominal

interception point. Figure 11 shows the tolerance ellipses including

95% of the interception points IPs (T0.95) for each experimental

condition (n = 12, 4 nBMD63 angles). We found that only 6 over

12 ellipses had statistically different eigenvalues (Table S5).

Moreover, the 95% confidence limits of the first eigenvector

never included the tangent in the mean IP to the trajectory of the

ball, deviating from it by .50u (Table S6). Therefore, the

individual responses were scattered randomly around the nIP,

consistent with the hypothesis that subjects were unable to fully

predict the invisible trajectory.

Nevertheless, subjects did take into account the theoretical

speed of the ball at arrival. In fact, the area of the ellipses increased

significantly (r2 = 0.848) with the speed of the ball at nIP (Fig. 6B),

similarly to Experiment 1. The slope of the linear regression was

Figure 7. Ball arrival time versus hand arrival time at the intersection point: data points for all single trials (n = 225 for eachcondition) and linear regression of Experiment 1 (red) and Experiment 2 (blue).doi:10.1371/journal.pone.0099837.g007

Figure 8. Paths of hand movement in a representative subject who performed both Experiment 1 (red) and Experiment 2 (blue). Allrepetitions (n = 15, tilt angle = 45u, nBMD = 610 ms) are plotted from the starting position to IP. Black dot represents nIP. 2D projections from aboveand from right side are plotted in A and B panels, respectively. Arrows indicate forward direction and upward direction of the movement in A and B,respectively.doi:10.1371/journal.pone.0099837.g008

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0.0035 m?s (95% confidence limits: [0.0024, 0.0044]), the

intercept was 0.0074 m2 (95% confidence limits: [0.0029, 0.0118]).

In contrast with Experiment 1, hitter intersection time was not

significantly correlated with ball intersection time (Fig. 7). For no

experimental condition was the correlation coefficient significantly

higher than zero (P.0.21); the slope of the regression averaged

across condition was 20.0257, and the average intercept was 0.66.

Therefore, if one considers the spatial distributions of IPs and the

lack of correlation between IT and ball intersection time

distributions, it appears that the where and when subjects placed

their hand to intersect ball trajectory were poorly related to each

other.

Movement characteristics. Typical paths of the hand are

plotted in blue in Fig. 8, and average movement parameters are

presented in Table S7. Hitting movements were systematically

straighter and of shorter duration than those in Experiment 1 (red

traces in Fig. 8). Average path curvature TCurv was 9.06 mm

(SD = 4.71 mm, n = 2661), significantly smaller than in Experi-

ment 1 (F1,28 = 20.307, P = 0.00011). Average movement duration

MD was 227.49 ms (SD = 61.24 ms), significantly shorter than in

Experiment 1 (F1,28 = 12.05, P,0.005). None of the movement

parameters depended significantly on any experimental factor (3-

way RM-ANOVA, 3 angles64 nBMDs615 repetitions, P.0.072),

except for the peak speed (PSpeed) and the speed in IP (Speed_IP)

which depended on nBMD (F3,42 = 6.183 and F3,42 = 14.51, both

P,0.001), being higher for longer nBMDs (higher target speeds).

In sum, the average interception position and time were roughly

close to those required by the instructions. Subjects tended to use

an interceptive strategy based on a global assessment of the unseen

ball kinematics across all tested conditions. A fine tuning of the

responses to the ball kinematics of single trials was limited, due to

the lack of on-line feedback about the critical terminal portion of

ball trajectory. In contrast with Experiment 1, the individual

responses were scattered randomly around the nIP, and hitter

intersection time was not significantly correlated with ball

intersection time. Nevertheless, there were indications of some

tuning of individual responses: the area of the 95%-tolerance

ellipses of the interception points increased significantly with the

speed of the ball at nIP, and the speed of arm movement was

significantly greater for faster balls than for slower balls, as in

Experiment 1. Learning involved spatial but not temporal

parameters of interception, presumably because participants

received feedback about the former but not the latter.

Overall, the results showed that subjects can still extrapolate

global aspects of gravitational motion of a target falling from an

incline even when the target stops moving before the fall, so that

there is no visual feedback about target kinematics during the free-

fall and no haptic feedback of hand-ball contact. In this

experiment, subjects were required to predict the expected time

of arrival of the ball at the nominal interception point. In the next

experiment, we tested whether subjects are also able to predict the

full trajectory of free-fall after the descent along the incline.

Experiment 3

In this experiment, the ball rolled along the incline until it was

stopped near the end of the track, as in Experiment 2, but here

participants were asked to draw with the hand in air the imaginary

trajectory of the ball, trying to mimic its motion beyond the stop.

They never saw the ball falling down the experimental incline

during or prior the experiment, nor were they told anything about

its possible motion. Mental extrapolation of the ball trajectory

could be based only on visual information obtained from the

previous rolling motion along the incline and on an internal model

of free-fall.

MethodsParticipants. Twelve subjects (9 females and 3 males, 2361

years old) participated in the experiment. None of them had

served in the previous experiments.

General procedures. The experimental setup and protocol

were similar to those of Experiment 2. In different blocks of trials,

the incline was tilted by 30u, 45u or 60u relative to the horizontal

(counterbalanced order). At each tilt angle, the ball was randomly

released from one of 4 different positions, resulting in different ball

motion durations nBMD (Table S8). These positions had been

chosen so that at least two of them for each angle were the same as

in Experiment 1 and 2, one was the same for all angles, and one

resulted in nearly the same ball speed (2.7 m?s21) at the lower end

of the incline for all angles.

As in Experiment 2, the ball rolled along the incline until it was

stopped by the rod near the end of the track. In the present

experiment, participants waited with the adducted, semi-flexed

Figure 9. Experiment 1. Speed profiles of hand movement of arepresentative subject (different from that of Fig. 8). The profile of thefirst repetition (red) is superimposed on the average (695% confidenceinterval, blue) over 4th to 15th repetitions of the same condition (tiltangle 45u, nBMD = 610 ms). Traces are aligned on the start time of ballmotion. nBMD is marked by the vertical line, and the hand arrival timeat IP in the 1st repetition by the cross.doi:10.1371/journal.pone.0099837.g009

Figure 10. Experiment 2. Effect of practice on spatial errors. Meanerror (695% confidence interval, over all subjects, tilt angles and nBMD,n = 180) is plotted for each repetition (red), together with theexponential best-fit (black), and the 95% decrement (dotted gray).doi:10.1371/journal.pone.0099837.g010

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upper limb, mid-pronated wrist. In this specific starting position,

the tip of the hand-held hitter rested at the stop-rod. Upon arrival

of the ball at the stop, participants had to immediately start tracing

with the hitter in air the imaginary trajectory of the ball, as if it

kept moving beyond the stop.

Data analysis. The kinematics of the hand-held hitter was

preprocessed as in the previous experiments. Next, we analyzed

both spatial and temporal aspects of the hand movements.

Spatial analysis. To compare trials with variable length of hand

movements, we considered only the portion of each trajectory

from the starting position to a final position at 12-cm Euclidean

distance from the starting position (all movements spanned at least

this distance), roughly comparable to the interception point in

Experiments 1 and 2. The trajectories were discretized in 100

equidistant points between starting position and final position. The

mean trajectory, along with the 95% confidence interval, was

computed over all trials for each condition (tilt angle and nBMD).

The same discretization was applied to the actual ball trajectory

during free-fall from the incline, as measured in separate

calibration trials (see Appendix). The deviation (TDe) of the

hand trajectory in single trials from the ball trajectory was

computed as the root mean square of the distances between each

pair of corresponding points on the two discretized trajectories.

The main direction TD of hand trajectory was defined as the slope

of the starting position-final position segment, and was compared

with the direction of actual ball trajectory defined in the same

manner.

Temporal analysis. In contrast with the relatively stereotyped hand

movements of the two previous experiments, movements were

more variable in this experiment. Thus, for each trial, the time of

onset of hand movements was derived according to an algorithm

recommended by Teasdale et al. [58] and Tresilian et al. [59].

The algorithm involves the following steps: a) the tangential

velocity (vT) is normalized to the maximum vmax: v = vT/vmax; b)

the sample k at which v first exceeds 0.1 is located; c) going back

from sample k, one stops at the first sample (m) for which v#0.09;

d) the standard deviation SDv of v is computed between sample m

and sample k; e) going back from sample m, one stops at the first

sample for which v # vm – SDv: this is the onset time (OT).

Statistics. Kolmogorov-Smirnov test showed that the trajec-

tory directions and deviations were not normally distributed (P,

0.001). Accordingly, for each tilt angle, the dependency of the

trajectory parameters on nBMD was assessed using Kruskal-Wallis

nonparametric test. Whenever a parameter did not depend

significantly on nBMD, statistics on its median and dispersion

were performed using Kruskal-Wallis and Ansari-Bradley tests

with tilt angle as factor (with Bonferroni correction). Statistics on

the characteristics of hand movements were performed using t-

statistics separately for each tilt angle and nBMD. Post-hoc

Bonferroni corrections were applied (P,0.05).

Results and DiscussionAlthough the participants never saw the ball falling off the

incline, on average they were able to draw its trajectory in air

reasonably well. The 95% confidence limits of the hand paths

computed over all repetitions of a representative subject are

compared with the actual ball trajectories for each condition in

Fig. 12. In half of the conditions (6/12), the confidence limits

included the actual ball trajectory, while in the other half the

confidence limits did not include the ball trajectory but the

discrepancy was limited.

Summary results from all participants are shown in Fig. 13. At

each tilt angle, the main direction TD of hand trajectory did not

depend significantly on nBMDs (all P.0.19). Instead, TD

depended significantly on tilt angle (P,0.001): the greater the

angle, the greater the TD. Median values of TD (n = 720 = 4

nBMD615 repetitions612 subjects) were 34.97u, 45.87u and

64.68u for 30u, 45u and 60u, respectively (Fig. 13A). Importantly,

these median values of TD were not significantly different from the

corresponding directions of actual ball trajectory (Kruskal-Wallis

Figure 11. Experiment 2. Spatial distribution of interception points. IPs in single trials (black dots) and 95%-tolerance ellipses (blue) for eachcondition. The first eigenvector with the 95%-confidence cone is drawn when significant. Same format as in Fig. 5.doi:10.1371/journal.pone.0099837.g011

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test, P.0.5), which were 36.13u, 47.65u, and 61.40u for 30u, 45uand 60u, respectively. This result confirms that, on average,

subjects were able to draw the trajectory in the correct direction.

However, as shown by Fig. 13A, there was considerable inter-trial

variability.

As a further test of the drawing accuracy, we derived a global

error estimate computed as the root mean square deviation (TDe)

of the discretized trajectory of the hand relative to the actual ball

trajectory (Fig. 13B). We found that, at each tilt angle, TDe was

small and did not depend significantly on nBMD (all P.0.40).

Instead, TDe depended on tilt angle (P,0.001). The greater the

angle, the smaller was the TDe: the median TDe was 1.46, 1.40

and 1.03 cm for 30u, 45u and 60u, respectively (the differences

between 60u and 30u and between 60u and 45u were significant

with Bonferroni test). Also the dispersion of TDe around the

median depended significantly on tilt angle (Ansari-Bradley test),

but did not depend on nBMD. The greater the angle, the smaller

was the dispersion of TDe (the differences between 60u and 30uand between 60u and 45u were significant after Bonferroni test).

Temporal movement characteristics. Average movement

parameters are presented in Table S9. For each condition, the

onset of hand movements (averaged over all subjects and

repetitions of each condition) occurred earlier than the arrival of

the ball at the stop (paired t-test, P,0.001), while the peak speed

occurred later than ball arrival at the stop (paired t-test, P,0.001).

The peak speed increased significantly (least-squares linear

Figure 12. Experiment 3. Frontal plane projection of 95%-confidence limits of hand paths (green, over all repetitions, n = 15) and actual ball path(orange) for each condition in a representative subject.doi:10.1371/journal.pone.0099837.g012

Figure 13. Experiment 3. Box-and-whisker plots of main direction TD of hand trajectory (A) and root mean square deviation TDe from the actualball trajectory (B). Bottom and top of the boxes correspond to the lower and upper quartile, respectively, and define the IQR. The intermediate line ineach box is the median. The lower and upper ends of the whiskers correspond to the smallest and largest datum within 1.5 IQR. Results from allnBMD, repetitions and subjects have been pooled for each condition (n = 720).doi:10.1371/journal.pone.0099837.g013

Interception of a Falling Ball

PLOS ONE | www.plosone.org 13 June 2014 | Volume 9 | Issue 6 | e99837

regression, r2 = 0.601, n = 180) with the speed of the ball at the

lower end of the incline. The slope of the linear regression across

all conditions was 0.2609 (95% confidence limits: [0.2294,

0.2924]), and the intercept was 1.271 m?s21 (95% confidence

limits: [1.177, 1.366]). Although the peak speed of hand

movements did not match that of actual ball motions, it was

substantially more modulated across conditions than in the

previous experiments. Thus, while the difference between maxi-

mum and minimum peak speed measured in different conditions

was 6.5 and 6.2% of the minimum value in Experiment 1 and 2,

respectively, the same difference was 41.6% in Experiment 3 (cf.

Tables S4, S7, S9).

In sum, subjects were able to mimic with hand movements

several features of unseen free-falls of the ball across a range of

different conditions. Average hand movements reproduced to a

good extent the path and overall direction of the actual ball

trajectory. Moreover, hand movements were performed with a

wide range of peak speeds with a trend similar to the range of ball

speeds across conditions. Subjects were more precise (less variable)

when they drew ball trajectories corresponding to greater tilt

angles.

General Discussion

We found evidence that participants took into account the

physics of fall from an inclined plane in all three experiments. Not

surprisingly, performance was most accurate when motion of the

ball was visible until interception and haptic feedback of hand-ball

contact was available (Experiment 1). Nevertheless, even when ball

motion was stopped at the end of the incline and participants

punched an imaginary moving ball (Experiment 2) or drew in air

the imaginary trajectory (Experiment 3), they were able to

extrapolate to some extent global aspects of the target motion,

including its path, speed and arrival time.

In Experiment 1, the ball underwent two different laws of

motion in successive phases of the descent. It first rolled down the

incline with a linear, roto-translatory motion whose kinematics

depended on the tilt angle and starting position. Afterwards, the

ball fell freely along a quasi-parabolic path which depended on the

exit velocity (modulus and direction) from the incline. The

deviation of the free-fall path relative to the previous rectilinear

path was appreciable for 30u and 45u tilts, but negligible for 60utilt. Ball acceleration was a fraction of 1g during the rolling phase,

whereas it was 1g during the free-fall phase at all tilts. Therefore,

there was no simple linear relationship between target motion

during free-fall and the previous rolling phase of the trajectory.

Nevertheless, the results showed that target motion was fully

extrapolated by the participants. At all tilt angles and starting

positions of the ball on the incline, the mean interception point

and time were very close to the ideal values, and the individual

movements in single trials tended to intersect the plane of ball

motion at points scattered along the path of fall (Fig. 5).

In this experiment, punching movements were most likely under

on-line visual control, as suggested by several indirect clues: the

distribution of interception positions along the ball path mentioned

above, the presence of significant curvature in hand movements in

the direction of target motion (Fig. 8), the robust coupling between

hand speed and target speed and an appreciable movement

duration (Table S4). Moreover, submovements were recognizable

in the tangential speed profile of the initial repetitions (Fig. 9). All

such behavioral hallmarks have previously been interpreted as

related to corrective interventions, mainly based on visual

feedback (see [11,60–64]).

Visual information, however, could not be used for on-line

adjustments of hand movements during the free-fall phase

preceding the interception, because the exit time of the ball from

the incline always occurred within the presumed ‘‘blind’’ period

corresponding to the visuomotor delay (e.g., [12,26,60]). In fact, it

is known that typical visuomotor delays for manual interceptions

range between 100 and 300 ms (e.g., [3,11–13]), whereas the

duration of free-fall from the exit time until nominal interception

time was only 18 to 80 ms, depending on the condition (see Table

S1). However, visual information and haptic feedback at ball-hand

contact (or the lack of it with a missed interception) could be used

to update the planning of movement for the next trials [65].

Extrapolation of target motion over the free-fall phase was

presumably based on the combination of multiple cues: visual

information obtained over the preceding rolling phase, feedback

and memory from previous trials, and an internal model of

physics. The combination of these cues resulted in an accurate

estimate of ball descent. The involvement of on-line visual

information during rolling has already been discussed. A memory

representation of target motion could be built from initial trials

using visual and haptic feedback as discussed above. Consistent

with this hypothesis, we found a rapid improvement of intercep-

tion timing (Fig. 3). However, rote memory alone was probably

insufficient to ensure a full representation of target motion during

free-fall, because the initial conditions of ball motion were

randomized across trials and therefore the exit velocity from the

incline was unpredictable. In particular, while tilt angle was

blocked (in counterbalanced order), the duration of ball motion

(nBMD) was randomized across trials.

We suggest that an internal model of target dynamics played a

key role in motion extrapolation. This putative internal model is

considerably more complex than that involved in the interception

of a target falling vertically with constant acceleration [8,23,25],

because ball motion in the present experiments depended on

several physical parameters, some invariant across trials (such as

the ball and surface properties, air drag), and others variable

across trials (such as the surface inclination and starting position on

the incline). A somewhat similar internal model, able to account

for complex acceleration profiles, has previously been suggested by

De Rugy et al. [44].

We did not monitor eye movements in our experiments, but a

previous study showed that naturalistic rolling objects elicit

effective tracking eye movements [43]. Thus, one may speculate

that also in our experiments the internal model of target motion

guided anticipatory eye movements along the expected trajectory,

and in turn the efference copy of eye movements guided the hand

to intercept the target at the expected position and time.

The potential role of the internal model of target motion was

even more crucial in Experiments 2 and 3, where the participants

saw the rolling phase of ball motion along the incline but did not

have any visual or haptic feedback about target kinematics during

the critical terminal part of the trajectory. Indeed, it has previously

been shown that the role of the internal model of gravity effects

becomes prevalent in the presence of visual occlusions over the

terminal part of a motion trajectory [66–68].

In Experiment 2, the average interception position and time

were close to those required by the instructions, but the individual

responses were tuned to the specific conditions of single trials to

only a limited extent. Indeed, there was no clear directional

distribution of the individual interception positions along the ball

path (Fig. 11). The observation that, in comparison with the hitting

movements of Experiment 1, those of Experiment 2 started later,

lasted less, were straighter, and did not exhibit obvious on-line

corrections speaks in favor of the idea that the latter movements

Interception of a Falling Ball

PLOS ONE | www.plosone.org 14 June 2014 | Volume 9 | Issue 6 | e99837

were more stereotyped and relied less on visual on-line informa-

tion. The bulk of the data from Experiment 2 suggests that subjects

used an interceptive strategy based on a global assessment of the

unseen ball kinematics across a range of different conditions, while

exploiting the residual visual information to maintain some degree

of hand-target motion coupling (as shown by the covariation of the

speed of arm movement with that of the falling ball). This

interpretation is in keeping with the previous suggestion that, in

the presence of visual occlusions over the terminal part of a motion

trajectory, subjects develop a new interception strategy specifically

adapted to the occluded protocol [68,69].

In Experiment 3, the subjects were able to draw in air hand

trajectories which roughly reproduced several features of the

actual ball trajectories across a range of different conditions, such

as ball path, direction and speed. Much previous work investigated

the nature of the internal representations of dynamics that are

associated with the observation of an object in motion (e.g.,

[18,21,40,48,70–72]). These representations vary widely depend-

ing on the context, yielding behavioral responses that are

compatible with naıve physics, geometric kinematics, simple

heuristics, or Newtonian physics (for a recent account, see [18]).

In general, when the visual presentation of the target is animated

(as opposed to a static picture) and the response required by the

observer is a motor action (as opposed to an explicit judgment or

verbal response), as in the case of our experiments, the responses

tend to be consistent with internalized Newtonian physics,

although the internal representations often model actual events

only in an approximate manner [21,26].

In conclusion, all 3 experiments showed that ball path and

kinematics generated by complex force patterns can be extrapo-

lated surprisingly well by the brain using both visual information

and internal models of target motion.

Appendix

Here we report the parameters of ball motion as derived from a

series of calibration trials performed before the experiments.

Characterization of Ball MotionEquations of motion on the inclined plane. The equation

of motion of a homogenous sphere rolling without slipping on a

tilted surface, under gravity and air drag linear in speed, is (see Fig.

S1 and Fig. S2B):

m

k€uu~{b _uuzmg sin að Þ{ mu

Rmg cos að Þ ðA1Þ

where m is the mass of the sphere, €uu is the linear acceleration of its

center of mass along the incline axis, b is the viscous friction

coefficient (it takes into account air viscosity, size and shape of the

ball moving through the air), _uu is the velocity of its center of mass

along the incline axis, g is the gravitational acceleration (9.81 m

s22), a is the angle of tilt of the incline relative to the horizontal, mu

is the coefficient of rolling resistance (unit of length), R is ball

radius.

The moment of inertia I of a sphere with the axis through the

center of mass is

I~2

5mR2 ðA2Þ

therefore

k~mR2

mR2zI~

5

7ðA3Þ

With

f muð Þ~mg sin að Þ{ mu

Rmg cos að Þ ðA4Þ

Equation A1 then reduces to

m

k€uu~{b _uuzf muð Þ ðA5Þ

If we integrate Equation A5 with initial conditions _uu0~0 and

u0~0, we obtain velocity:

_uu tð Þ~f muð Þb

1{e{kbmt

� �ðA6Þ

and position:

u tð Þ~f muð Þb

t{m

kb1{e{k

bmt

� �� �ðA7Þ

Characterization of the incline. Given the above expres-

sions for speed and position (Eq. A6 and A7), a full characteriza-

tion of ball motion on the incline can be achieved once b and mu

are known. b and mu were evaluated with a procedure based on the

analysis of digital images of ball trajectories recorded at 100 Hz by

a Digital Video Camera (Sony Handycam HDR-SR8E). These

parameters were estimated experimentally by analyzing 36 video

recordings of ball rolling motion (spatial resolution: 192061080

pixels, temporal resolution: 100 interlaced fields per second) on the

incline (between the releasing position and the lower end of the

incline) inclined at 45u. We performed 3 video recordings for each

of the 12 releasing positions of Experiment 1 and 2. The beginning

of video recording and that of ball motion were not synchronous.

Therefore, we identified the onset frame of video-recorded ball

motion as the first video frame at which the ball center of mass

exceeded the threshold velocity of 0.1 m?s21. The ball position in

this frame is indicated as u0, and ball speed as V0.

Motion equation for video recorded ball motion was obtained

integrating twice Equation A5 with initial conditions _uu0~V0 and

u t~0ð Þ~u0:

u tð Þ~

f muð Þb

t{m

kb1{e{k

bmt

� �� �zV0

m

kb1{e{k

bmt

� �zu0

ðA8Þ

The parameters b and mn were obtained by (nonlinear least-

squares) best-fitting Equation A8 to the ball kinematics recorded in

the videos from the onset frame of video-recorded ball motion and

the end of motion recording. The estimated value of b was b= 1.77?1025 (61.79 1025 SD, n = 36) and the estimated value of

mu was mu = 2.2 1023 (65.17?1024 SD).

Ball acceleration was roughly constant on the plane for a given

inclination. It was 3.21 (60.25?1023 SD) m?s22, 4.71 (60.39?1023

Interception of a Falling Ball

PLOS ONE | www.plosone.org 15 June 2014 | Volume 9 | Issue 6 | e99837

SD) m?s22 and 5.90 (60.50?1023 SD) m?s22 for tilt angles of to

30u, 45u and 60u, respectively (mean and SD was computed for the

longest ball motion duration on the plane with 1 ms time step).

Ball motion in air. The equation of motion of a sphere in air

(i.e. after exiting from the lower end of the incline), under gravity

and air drag linear in speed, is (see Fig. S2C):

m€xx~{b _xx ðA9Þ

m€yy~{b _yyzmg ðA10Þ

where €xx (€yy) is the linear acceleration of its center of mass along the

horizontal (vertical) axis and _xx ( _yy) is the velocity of its center of

mass along the horizontal (vertical) axis (Fig. S1).

The corresponding velocity profiles are:

_xx~ _xx0e{bmt ðA11Þ

_yy~mg

bz _yy0z

mg

b

� �e{

bmt ðA12Þ

and positions are:

x tð Þ~ m

b_xx0z 1{e{

bmt

� �ðA13Þ

y tð Þ~{mg

btz

m

b_yy0z

mg

b

� �1{e{

bmt

� �ðA14Þ

Solutions were obtained by integrating Equation A9 and A10

with initial conditions, x0~0, y0~0 and t0~0. In the above

equations, the point x0,y0ð Þ~ 0,0ð Þ represents the position of the

center of mass of the ball when it exits from the incline.

We verified that edge effects were negligible in the experimental

conditions by means of measurements of the ball speed at the

lower end of the incline and of the flight time of the ball in air

through the high resolution video-camera and the 3-axial

accelerometer placed at 1 ball-radius distance below and to the

right from the nominal interception point.

Comparison of Different Laws of MotionFigure S2 (right panels) plots the isochronous lines correspond-

ing to falls under different conditions. Different colors identify

different durations of fall (see Figure legend). Each isochronous

line depicts the locus of starting positions yielding constant

duration of fall along planes of different inclinations. For sliding

motion under gravity on a friction-less plane, the isochronous lines

are circles passing through the end of the planes (Fig. S2A). For

rolling motion without slipping under gravity and air drag linear in

speed (Eq. A1) and for free-fall after rolling on the incline, the

isochronous lines become egg-like (Fig. S2B–C).

Supporting Information

Figure S1 Schematic of ball motion. The ball is illustrated

in 3 different positions: while rolling down the incline, at the exit of

the incline, and at the nominal interception point (at distance d1

from the incline exit).

(TIF)

Figure S2 Schematic illustration of different types offalls from an inclined plane. A. Sliding of a parallelepiped

under gravity on a friction-less plane. B. Rolling without slipping

of a sphere under gravity and air drag linear in speed. C. As in B,

followed by free-fall under gravity and air drag. Left panels: free-

body diagrams of falls. a: inclination angle relative to the

horizontal. FN: ground reaction force. Fg: gravitational force. Fb:

air resistance force. Fs: sliding resistance. Mv: rolling resistance

moment. In C, d1 corresponds to the arrival point after free-fall (as

in Fig. S1). Right panels: isochronous lines for falls at different tilt

angles. Each isoline connects the x,y starting positions yielding the

same duration of fall along planes of different inclinations: red for

550 ms, cyan for 610 ms, green for 670 ms, and purple for

730 ms. In A, the isolines corresponding to 670 and 730 ms are

not drawn because out-of-scale for higher tilt angles. In C, the

isolines have been computed for d1 = 0.15 m.

(TIF)

Table S1 Ball motion parameters in Experiment 1(Incline and Air) and Experiment 2 (Incline).

(DOCX)

Table S2 Square-root of the eigenvalues of 95% toler-ance ellipses in Experiment 1. They correspond to the semi-

axes of the ellipses (cm). *Eigenvalues not statistically distinct.

(DOCX)

Table S3 Orientation (in degrees) of the major axis of95% tolerance ellipses in Experiment 1.

(DOCX)

Table S4 Mean values and standard deviations (SD) ofthe kinematical variables MD, ISpeed, PSpeed, Spee-d_IP and TCurv as a function of the four nBMDs and thethree incline tilting angles in Experiment 1.

(DOCX)

Table S5 Square-root of the eigenvalues of 95% toler-ance ellipses in Experiment 2. They correspond to the semi-

axes of the ellipses (cm). *Eigenvalues not statistically distinct.

(DOCX)

Table S6 Inclination (in degree) of the major axis of95% tolerance ellipses in Experiment 2.

(DOCX)

Table S7 Mean values and standard deviations (SD) ofthe kinematical variables MD, ISpeed, PSpeed, Spee-d_IP and TCurv as a function of the four nBMDs and thethree incline tilting angles in Experiment 2.

(DOCX)

Table S8 Ball motion parameters in Experiment 3.

(DOCX)

Table S9 Experiment 3. Mean values and standard deviations

(SD) of the kinematical variables: time interval between OT and

ball stop time (BST), PSpeed, Interval time between PSpeed time

and BST as a function of the four ball motion duration for each

incline tilt and the three incline tilting angles. Note that ball

motion duration time1, time2, time3, time4 were not the same for

the three incline tilting angles.

(DOCX)

Interception of a Falling Ball

PLOS ONE | www.plosone.org 16 June 2014 | Volume 9 | Issue 6 | e99837

Acknowledgments

We thank Emanuele Giacomozzi for help with the setup.

Author Contributions

Conceived and designed the experiments: BLS MZ FL. Performed the

experiments: BLS MZ. Analyzed the data: BLS MZ. Contributed to the

writing of the manuscript: BLS MZ FL.

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